The present invention is directed to a method and system for estimating the number of emitters having signals received by an array of sensors, and more specifically to an improved method and system for estimating the number of emitters having wavefronts impinging on an antenna array that may be used to improve existing systems that determine various characteristics of those wavefronts.
The identification of the characteristics (such as direction of arrival, strength, frequency, etc.) of wavefronts impinging on an array of sensors is of importance in a wide variety of applications, including radio direction finding, radar, sonar, surveillance, medical imaging, geophysics, etc. Several methods and systems have been developed to identify these characteristics, with the most promising relying on the eigenstructure approach developed by R. O. Schmidt in his Ph.D. thesis entitled "A Signal Subspace Approach to Multiple Emitter Location and Spectral Estimation", Stanford University, 1981 (the method therein is known as MUSIC). MUSIC has been changed and updated (see, for example, the ESPRIT algorithm discussed in U.S. Pat. No. 4,965,732 issued to Roy, et al. on Oct. 23, 1990) and many other methods and systems are known.
The need to accurately estimate of the number of emitters having wavefronts impinging on the sensor array is critical to the accurate estimate of wavefront characteristics in all of the above-identified methods and systems. Prior art methods and systems that rely on eigenvalues to determine the number of emitters include the Minimum Description Length (MDL) and Akaike Information Criterion (AIC) methods, and the Williams and Johnson Sphericity Test. Nevertheless, the prior art methods and systems typically overestimate the number of wavefront emitters, sometimes quite badly. As will be discussed in more detail below, one of the problems associated with the prior art estimation of the number of emitters is that they are not robust to relaxing the assumption that noise is uncorrelated and of equal power at each sensor. That is, they do not handle correlated, unequal-level noise as well as uncorrelated, equal-level noise and, as a result, overestimate the number of emitters.
The operation of the prior art methods and systems for determining the characteristics of wavefronts is beyond the scope of the present invention and will not be discussed herein as those methods and systems are known or available through other sources to ones of skill in the art. However, a brief description of a prior art technique for estimating the number of emitters follows to assist in understanding the present invention.
The response of an array of m sensors to a unit-energy plane wavefront arriving from the direction .theta. is a complex m.times.1 vector, typically identified as a steering or direction vector: EQU a.sub..theta. =(a.sub.1, a.sub.2, . . . , a.sub.m).sup.T ( 1)
where .sup.T indicates the transpose.
When n signals, x.sub.1 [t], x.sub.2 [t], . . . , x.sub.n [t], are carried on planewaves with the same wavelength, but different directions, into the sensor array, the superimposed response of the array to the planewaves is ##EQU1## where n[t] is the vector of noise received across the array from the sensor system and background;
A=[a.sub.1, a.sub.2, . . . , a.sub.n ]; and
x=(x.sub.1 [t], x.sub.2 [t], . . . , x.sub.n [t]).sup.T.
The array sample at each time t, y[t], is identified herein as a snapshot. It is assumed that the signals x.sub.k [t] change slowly enough so that, at any point in time, their value is simultaneously equal across the array. The sensor correlation matrix E(y[t]y*[t]) is denoted by R, and the signal correlation matrix E(x[t]x*[t]) by S, both being stationary in time. The prior art methods and systems assume that the noise is independent in time, uncorrelated across sensors, and of equal power at each sensor, with covariance matrix EQU Q=E(n[t]n*[t])=.sigma..sup.2 I (3)
In these terms, it follows that R has a linear structure given by EQU R=ASA*+.sigma..sup.2 I (4)
with eigenvalues 0.ltoreq..lambda..sub.1 .ltoreq..lambda..sub.2 .ltoreq. . . . .ltoreq..lambda..sub.m.
From the linear structure of equation (4) it is apparent that those vectors orthogonal to the columns of A are eigenvectors with eigenvalues all equal to the noise power .sigma..sup.2. For n signals, there will be m-n of these noise eigenvalues. If there is only one planewave signal present of power .sigma..sub.1.sup.2, there are m-1 noise eigenvalues and one signal eigenvalue equal to the signal power .sigma..sub.1.sup.2. The corresponding signal eigenvector is the steering vector. However, this pattern does not continue for more than one signal. For uncorrelated signals EQU S=diag(.sigma..sub.1.sup.2, . . . , .sigma..sub.n.sup.2) (5)
and the signal eigenvectors are not the direction vectors, but they do span the same linear subspace. (This is the basis of MUSIC.) The average of the eigenvalues is the total signal and noise power as set forth below: ##EQU2## or, in other words, ##EQU3##
To prove this, using the structure of R as in equation (4), note that (for A* defined as the complex conjugate transpose of the matrix A), ##EQU4## and that the steering vectors (columns of A) ideally have constant norm-squared m.
The relationship of equation (6) means that the arithmetic mean (AM) of the eigenvalues will be the same for any array, of any number of elements, receiving signals of these powers from any directions. The geometry does not affect the AM(.lambda..sub.1, . . . , .lambda..sub.m).
With reference to the MDL (Minimum Description Length) and AIC (Akaike Information Criterion) methods, it is noted that both use only the number of snapshots N and the eigenvalues .lambda..sub.1, . . . , .lambda..sub.m of the sensor correlation matrix R. Both MDL and AIC take the number of emitters n as the value of m-k for which the flatness (or levelness) of the first k "noise" eigenvalues, plus a correction term, is minimum. The flatness measure is basically the ratio of the geometric mean (GM) to the arithmetic mean (AM) of the .lambda.'s. The arithmetic mean is the usual average. The geometric mean is the length of the side of a k-dim cube with volume equal to a k-dim box with sides .lambda..sub.j. Therefore 0.ltoreq.GM/AM.ltoreq.1, since the eigenvalues are not negative. The ratio GM/AM equals 1 only when the graph of the eigenvalues is flat. The closer the ratio is to 1, the more nearly equal the k numbers are. MDL and AIC (and the Williams Johnson Sphericity Test) evaluate the flatness of the first k eigenvalues. They basically use the logarithm of (GM/AM).sup.k for the flatness f; that is ##EQU5##
MDL adds to this the correction term ##EQU6## and sets n=m-k when the flatness plus the correction is minimum.
Interpreting f as flatness helps to explain why it is not a robust statistic. For example, consider the flatness of n numbers all equal to 1, except for one equal to x. Before the log, it is ##EQU7##
With reference to FIG. 1, for large x (as for a signal eigenvalue k well above the noise level .sigma..sup.2 =1), the flatness f will tend to go down from one toward zero. This is good, for it says that this flatness measure will catch sudden jumps above the noise floor. However, for small x (as x.fwdarw.0), the flatness will also tend toward 0. This is not good, because it means that when the noise eigenvalues are increasing (due to correlated noise) our flatness measure will tend to think the noise eigenvalues are not very flat. Either way, the flatness slides off the peak at x=1.
By way of further explanation, the spectrum of five eigenvalues (for five sensors, m=5) in the presence of uncorrelated noise may be seen in FIG. 2. In this example, there are two signal eigenvalues (values 4 and 5) and three noise eigenvalues (values 1-3). The three noise eigenvalues are relatively flat and are thus distinguishable from the signal eigenvalues. Prior art methods and systems would likely be able to correctly estimate the presence of the two signal sources. They work roughly as follows: assuming the first eigenvalue is a noise eigenvalue, they continue to add one more candidate noise eigenvalue and test for flatness f until the test indicates they are no longer flat. All remaining eigenvalues are assumed to be signal related. This procedure is usually adequate when the eigenvalues are as illustrated in FIG. 2. However, in the presence of correlated noise, the noise eigenvalues are not as flat, as may be seen by reference to FIG. 3. As seen therein, the prior art would look at the first three eigenvalues only and the third noise eigenvalue would likely not pass the flatness test. The test would likely indicate that only the first two eigenvalues are noise related, and, as a result, the number of emitters would be overestimated by one (that is, the test would indicate the presence of three emitters, not two).
The prior art techniques fail because they are, in effect, near-sighted. The problem with this is that the flatness test is a microscope; it only sees the noise eigenvalues at hand, and does not see them in relation to the rest of the eigenvalues. Thus, we are easily able to see that the first three eigenvalues in FIG. 3 are essentially flat when viewed in the context of all five eigenvalues. However, the prior art techniques are not aware of the two signal eigenvalues as they evaluate the flatness of the three noise eigenvalues. Under the microscope of the prior art techniques, the three noise eigenvalues, perturbed as a result of coherent noise, are given a poor intrinsic flatness rating, and the test declares that only the first two locally flat eigenvalues are noise. The prior art techniques resist including the third noise eigenvalue, preferring to stick with just the first two. Thus the noise count is low, producing high signal counts.
Accordingly, it is an object of the present invention to provide a novel method and system for estimating the number of emitters that obviates the problems of the prior art.
It is a further object of the present invention to provide a novel method and system for estimating the number of emitters in which the sensitivity to eigenvalue flatness is reduced by increasing the eigenvalues.
These and many other objects and advantages of the present invention will be readily apparent to one skilled in the art to which the invention pertains from a perusal of the claims, the appended drawings, and the following detailed description of preferred embodiments.